Finding Zeros of x³ - 2x² - 11x + 12

Algebra 2 · Axiom Academy

EXAMPLE Finding the Zeros of f(x) = x^3 - 2x^2 - 11x + 12 Complete process using the Rational Root Theorem, synthetic division, and factoring Find all zeros of f(x) = x^3 - 2x^2 - 11x + 12 . Excellent work! You've successfully found all zeros of a cubic polynomial. Here's what we learned: Rational Root Theorem — gives us a list of possible rational zeros to test (factors of the constant term ÷ factors of the leading coefficient). Synthetic Division — an efficient way to test a potential zero and find the quotient polynomial in one pass. Factoring Strategy — once one zero is found, factor out that linear term and keep working with the simpler quotient. Complete Solution — for this cubic, all three zeros were rational: x = 1, 4, -3 . Double-check — substituting each zero back into f gives f(1) = f(4) = f(-3) = 0 , and (x-1)(x-4)(x+3) expands back to the original polynomial exactly. This systematic approach works for any polynomial: start with the Rational Root Theorem, test candidates with synthetic division, then factor whatever remains.

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